Proper Orthogonal Decomposition Extensions and Their Applications in Steady Aerodynamics

107 Pages·2003·2.18 MB·English

Proper Orthogonal Decomposition Extensions
and Their Applications in Steady Aerodynamics
by
Bui Thanh Tan
B.Eng. in Aeronautics
Ho Chi Minh City University of Technology, Vietnam, 2001
SUBMITTED TO THE SMA OFFICE IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF ENGINEERING
IN HIGH PERFORMANCE COMPUTATION FOR
ENGINEERED SYSTEMS (HPCES)
AT THE
SINGAPORE-MIT ALLIANCE
2003
Signature of author:...................................................
HPCES Programme
Certified by:............................................................
Assoc. Prof. Murali Damodaran
SMA Fellow, NTU
Thesis Advisor
Certified by:............................................................
Assist. Prof. Karen Willcox
SMA Fellow, MIT
Thesis Advisor
Accepted by:...........................................................
Assoc. Prof. Khoo Boo Cheong
Programme Co-Chair
HPCES Programme
Accepted by:...........................................................
Prof. Jaime Peraire
Programme Co-Chair
HPCES Programme
(cid:1)c 2003 Singapore-MIT Alliance. All rights reserved.
Proper Orthogonal Decomposition Extensions
and Their Applications in Steady Aerodynamics
by
Bui Thanh Tan
In Partial Fulfillment of the Requirements for the
Degree of Master of Engineering in High Performance
Computation for Engineered Systems (HPCES)
ABSTRACT
The development and application of model reduction techniques have been
widelyinvestigatedforproblemsinunsteadyaerodynamicsystems,drivenmainly
by the need to develop effective economical computational models capturing as
much flow physics as possible. These techniques which have been based on a
number of approaches such as balanced truncation, Arnoldi, proper orthogonal
decomposition etc are designed to obtain a reduced-order model for both linear
and nonlinear systems. However there has been a paucity in the application of
these techniques in the area of steady aerodynamics. The work reported in this
thesisdemonstrateshowtheproperorthogonaldecomposition(POD)technique
can be extended to a number of applications in steady aerodynamics.
In the first instance the POD approach is coupled with a cubic spline in-
terpolation procedure to develop reliable fast, low-order models for accurately
predictingsteadyaerodynamicflowfieldsforarbitraryvaluesorvariationinflow
parameters such as the angle of attack or inflow Mach number. Results on the
prediction of steady transonic aerodynamic flowfield solution past an airfoil at
arbitrary values of angle of attack or Mach number show that accurate flow-
field predictions can be obtained, including cases that were not sampled in the
ensemble of snapshots.
The second extension concerns a “gappy” POD technique for the recon-
structionofincompleteorinaccurateaerodynamicflowfielddata. Thefirstcase
corresponds to the complete reconstruction of pressure field around an airfoil
from the knowledge of pressure data defined only at the airfoil surfaces. The
second case corresponds to the one in which POD snapshots are reconstructed
from an incomplete set of aerodynamic data. Gappy POD is shown to be an
effective technique for reconstruction of complete aerodynamic flow field data
fromlimitedmeasurementsorincompletedata. Thisapproachdemonstratesan
effective way in which experimental and computational aerodynamic data can
be combined to predicate accurate aerodynamic flow fields cheaply.
Finally it is shown how an extension of the gappy POD can be used to
formulate and cast inverse airfoil shape design and flowfield prediction for an
arbitrary airfoil problems, in which steady aerodynamic plays a role, as gappy
data problems. The extension of the methodology to constrained airfoil shape
designproblemisalsodiscussed. Thegappydesignmethodologyhasbeenshown
to be a simple, effective and an efficient airfoil shape design technique.
Thesis Advisors:
1. Assoc. Prof. Murali Damodaran, SMA Fellow, NTU
2. Assist. Prof. Karen Willcox, SMA Fellow, MIT
ACKNOWLEDGMENT
It is a great pleasure to thank my first advisor, Prof. Damodaran Murali,
for training me in computational fluid dynamics and my second advisor, Prof.
Karen Willcox, for sharing with me her deep knowledge in proper orthogonal
decomposition.
I would also like to thank Prof. Nguyen Thien Tong and Prof. Nhan Phan
Thien for encouraging me to study under the Singapore-MIT Alliance (SMA).
Without their guidance, I would not have accomplished this work.
IamverypleasedtothankallofmyfriendsfromVietnam,Malaysia,China,
India and Singapore from the third batch of High Performance Computation
for Engineered Systems (SMA-HPCES 2001) for their help and encouragement
which made my life in SMA so interesting that it will remain etched in my
memory forever.
This work has been supported through a SMA scholarship and I would like
to thank the Singapore Government for creating this scholarship for students
aroundtheworld. IamsogratefultoallProfessorsintheSMA-HPCESprogram
for teaching me many useful subjects for my research work.
Finally, I would like to thank my family for their continual support and
encouragement during my research.
DEDICATION
To my mom
H`a Th.i Maˆ.n
my wife
P.T. Coˆng Minh
and my brother
Hˆo` B˜ao Hˆan
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Compressible Aerodynamic Flow Simulation 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Flow Model for Inviscid Flow . . . . . . . . . . . . . . . . . . . . 13
2.3 Finite Volume Formulation . . . . . . . . . . . . . . . . . . . . . 13
2.4 Time Stepping Scheme . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Convergence Acceleration Strategies . . . . . . . . . . . . . . . . 17
2.5.1 Enthalpy Damping . . . . . . . . . . . . . . . . . . . . . . 17
2.5.2 Implicit Residual Smoothing . . . . . . . . . . . . . . . . 18
2.5.3 Multigrid Strategies . . . . . . . . . . . . . . . . . . . . . 19
2.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Proper Orthogonal Decomposition Extensions 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Proper Orthogonal Decomposition (POD) . . . . . . . . . . . . . 25
3.3 POD with Interpolation (PODI) . . . . . . . . . . . . . . . . . . 27
3.4 POD for Reconstruction of Missing Data (Gappy POD) . . . . . 29
3.5 POD with an Incomplete Snapshot Set (Iterative gappy POD) . 31
v
4 Application Problems in Steady Aerodynamics Using POD Ex-
tensions 33
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Parametric Applications in Steady Transonic Aerodynamics . . . 34
4.3 Reconstruction of Missing Aerodynamic Data . . . . . . . . . . . 35
4.4 Airfoil Design Optimization . . . . . . . . . . . . . . . . . . . . . 36
4.4.1 Inverse Design Using the Gappy POD Method . . . . . . 36
4.4.2 Constrained Airfoil Design Optimization . . . . . . . . . . 38
4.5 Prediction of Flowfield for an Arbitrary Airfoil . . . . . . . . . . 40
5 Results and Discussion 41
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Parametric Applications in Transonic Aerodynamics . . . . . . . 42
5.2.1 Flowfield Prediction at an Arbitrary Angle of Attack . . . 42
5.2.2 FlowfieldPredictionforanArbitraryPairofAngleofAt-
tack and Mach Number . . . . . . . . . . . . . . . . . . . 44
5.2.3 Estimation of the Greatest Lift Coefficient for a Given
Angle of Attack Range . . . . . . . . . . . . . . . . . . . . 52
5.2.4 TrackingtheAngleofAttackforaGivenFlowfieldAround
an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 Reconstruction of Missing Aerodynamic Data . . . . . . . . . . . 54
5.3.1 Reconstruction of Aerodynamic Flowfield Data . . . . . . 54
5.3.2 Incomplete Flowfield Snapshot Set . . . . . . . . . . . . . 58
5.4 Airfoil Design Optimization . . . . . . . . . . . . . . . . . . . . . 64
5.4.1 Inverse Design Using Gappy POD Method. . . . . . . . . 64
5.4.2 Constrained Airfoil Design Optimization . . . . . . . . . . 78
5.5 Prediction of Flowfield around an Arbitrary Airfoil . . . . . . . . 82
5.5.1 Grid Generation Around an Arbitrary Airfoil . . . . . . . 82
5.5.2 Prediction of Flowfield around an Arbitrary Airfoil . . . . 85
6 Conclusions and Recommendations 89
vi
List of Figures
2.1 Structured C-grid for finite volume scheme. . . . . . . . . . . . . 13
2.2 Discretization in space . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Theconvergenceofthenumber ofsupersonicpoints andresidual
for the NACA 0012 with M = 0.8 and AOA = 0.45 . . . . . . . . 22
2.4 Theconvergenceofliftanddragcoefficientsandentropycontours
for the NACA 0012 with M = 0.8 and AOA = 0.45 . . . . . . . . 23
2.5 Pressure contours and pressure coefficients for the NACA 0012
with M = 0.8 and AOA = 0.45 . . . . . . . . . . . . . . . . . . . 23
5.1 Comparisonofpredictedpressurecontours(dash)andCFDpres-
surecontours(solid)foraMachnumberof0.8andangleofattack
of 0.77◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 The convergence of lift coefficient versus the number of POD
modes for a Mach number of 0.8 and angle of attack of 0.77◦. . . 44
5.3 Comparisonofpredictedpressurecontours(dash)andexactpres-
surecontours(solid)foraMachnumberof0.8andangleofattack
of 0.45◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 Comparisonofpredictedupperpressurecoefficients(square),pre-
dicted lower pressure coefficients (circle), exact upper pressure
coefficients (dash) and exact lower pressure coefficients (solid)
forM=0.8andAOA=0.45◦. Cp∗ isthesonicpressurecoefficient. 47
5.5 Variation of percentage error versus the number of POD modes
in log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
vii
5.6 Comparisonofpredictedpressurecontours(dash)andexactpres-
sure contours (solid) for a Mach number of 0.812 and angle of
attack of 0.5◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.7 Comparisonofpredictedpressurecontours(dash)andexactpres-
sure contours (solid) for a Mach number of 0.812 and angle of
attack of 1.1◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.8 The reconstruction of the pressure field from the airfoil surface
pressuredistribution(dash),comparedwiththeoriginalcontours
(solid). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.9 The 2-norm of the pressure reconstruction error versus the num-
ber of POD modes. For all four cases, the pressure measurement
points were equally spaced over the airfoil surface. . . . . . . . . 57
5.10 Reconstructionofthesecondsnapshot(dash),comparedwiththe
original contours (solid). . . . . . . . . . . . . . . . . . . . . . . . 60
5.11 Reconstruction of the 23rd snapshot (dash), compared with the
original contours (solid). . . . . . . . . . . . . . . . . . . . . . . . 61
5.12 The eigenvalue spectrum for construction of the POD basis from
an incomplete snapshot set. Shown are the POD eigenvalues at
various stages in the iterative process. . . . . . . . . . . . . . . . 62
5.13 Construction of POD modes from an incomplete snapshot set.. . 63
5.14 The airfoil snapshots from baseline RAE 2822 (dash) and the
Korn airfoil (solid) and corresponding POD eigenvalues. . . . . . 66
5.15 Korn airfoil design. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.16 The design (dash) airfoils from 63 RAE 2822 based snapshots. . 71
5.17 Thedesignairfoils(dash)airfoilsfrom90snapshotswithbaseline
RAE 2822 and NACA 64A410. . . . . . . . . . . . . . . . . . . . 72
5.18 The design (dash) airfoils from restarted gappy POD method. . . 74
5.19 The exact Korn (solid) and the design (dash) airfoils from 63
RAE based airfoils in transonic regime. . . . . . . . . . . . . . . 76
5.20 The exact Korn (solid) and design (dash) airfoils from new en-
sembleofsnapshotsbasedontheintermediateairfoilintransonic
flow (restart). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
viii
5.21 The initial airfoil (solid) and the design airfoil with five POD
modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.22 The pressure coefficients comparison for the design airfoil with
five POD modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.23 Theinitialairfoil(solid)andthedesignairfoilwithtenPODmodes. 81
5.24 The pressure coefficients comparison for the design airfoil with
ten POD modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.25 The grid generation with 29 POD modes for Korn airfoil from
RAE based snapshots. . . . . . . . . . . . . . . . . . . . . . . . . 83
5.26 Thegridgenerationwith29PODmodesforGOE117airfoilfrom
RAE based snapshots. . . . . . . . . . . . . . . . . . . . . . . . . 84
5.27 Thepressurefieldpredictionwith29PODmodesforKornairfoil
from RAE based snapshots. . . . . . . . . . . . . . . . . . . . . . 87
5.28 The pressure field predictions for NACA 63212, HQ 2010 and
GOE 117 airfoils from RAE based snapshots. . . . . . . . . . . . 88
ix
List of Tables
5.1 ThetoleranceandcorrespondingresultbyGoldenSearchmethod
for greatest lift coefficient problem. . . . . . . . . . . . . . . . . . 53
5.2 ThetoleranceandcorrespondingresultbyGoldenSearchmethod
for the angle of attack tracking problem. . . . . . . . . . . . . . . 54
5.3 Optimal cost versus number of POD modes for the subsonic in-
verse design of Korn airfoil. . . . . . . . . . . . . . . . . . . . . . 69
5.4 Optimal cost versus number of POD modes for subsonic inverse
design cases of NACA 63212, HQ 2010 and GOE 117 airfoils. . . 69
5.5 The lift and drag coeficient for the initial and optimal airfoils,
five POD modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.6 The lift and drag coeficient for the initial and optimal airfoils,
ten POD modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
x